Problem: What is the slope of the line tangent to $f(x) = x^{2}-2x-7$ at $x = 4$ ?
The slope of the tangent line is $ \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{((x+\Delta x)^{2}-2(x+\Delta x)-7) - (x^{2}-2x-7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{(x^{2}+2x \Delta x+\Delta x^{2}-2(x+\Delta x)-7) - (x^{2}-2x-7)}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{x^{2}+2(x \Delta x)+\Delta x^{2}-2x-2(\Delta x)-7-x^{2}+2x+7}{\Delta x}$ $ = \lim_{\Delta x \to 0} \frac{2(x \Delta x)+\Delta x^{2}-2(\Delta x)}{\Delta x}$ $ = \lim_{\Delta x \to 0} 2x+\Delta x-2$ $ = 2x-2$ $ = (2)(4)-2$ $ = 6$